3 Answers
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The JEPD theory was developed in three stages, which I summarized in my answer to
What was the reasons for documentary hypothesis?. It isn't complete nonsense; it's the culmination of several centuries of Bible scholarship. Whether it is a giant leap forward or a brilliant mistake, I'm not certain.

On the one hand, it's rational, and it fits different types of writing into neat categories. It has a ready answer for things that might be hard to explain otherwise.

Perhaps the most serious scholarly argument against the JEPD hypothesis is that it doesn't explain why a later editor would bring all these sources together, if the traditions had kept them separate. The more likely scenario, according to this argument, is that the doublets and stylistic differences were part of a tradition that began with Moses and was accumulated over the years.

Deuteronomy describes the death and burial of Moses (the book's supposed author). Moses could have finished the book before ascending the mountain, but that seems much less likely than a contribution from another author to complete it.

Numbers 12:3 Now Moses was a humble man: the most humble man in all the earth. It seems to me that humble man would have a hard time writing that sentence. This again indicates an additional author, or at least editor.

Neither of these rule out Moses as the primary author of all 5 books, though, especially considering the necessity of the time of frequently hand-copying the books for preservation. Addendums had a habit of finding their way into the original text over time.

The basic premise of the Prolegomena is the identification of certain contradictions throughout the Pentateuch that point toward a progression toward the "Judaism" of the Exilic and second Temple period. The arguments are very nuanced, but Wellhausen's attention to detail is staggering. Central to his argument is the apprent change of opinion regarding the centrality of Jerusalem for the worship of Yahweh. The earlier strata seem unconcerned by, for instance, Solomon (or Samuel, Elisha, etc.) making sacrifices to God outside of Jerusalem, which is strictly forbidden in the later D and P sources. Moreover, none of the above mentioned people were Levites who were the sole practitioners of the temple cult.

Biblical scholarship has come a long way since Wellhausen, but one would be remiss to toss out his work as passe or otherwise "disproven." The DH is broadly accepted by nearly every serious scholar of the Hebrew Bible (with modification and individual nuance, of course) for good reason.

Don't accept (or reject) the theory just because someone on the internet said so; read the first few chapters of the Prolegomena (with a Hebrew Bible next to you if you can)--the man was truly brilliant.

Post Script: Incidentally, Wellhausen was a devout Christian--keep that in mind when you read about his theories.